Since a $\bar{\partial}$-Operator is an elliptic Operator, you can use elliptic theory in order to prove the Serre duality. In fact the Serre duality is a kind of corollary of the"fundamental theorem" (as I Know it). In fact, you do not need the Hodge theorem, since the Hodge theorem itself is a corollary of the theorem.

For a reference of this "fundamental theorem" (perhaps slightly reformulated) I would refer to one of the following:

I thought I'd offer a high-tech alternative for certain varieties. If $X$ is smooth and projective over a field $k$ then Bondal and van den Bergh give a proof here that $D^b(\mathrm{Coh}X)$ is saturated which is a strong representability condition on cohomological/homological functors to the category of $k$ vector spaces. It follows immediately that $D^b(\mathrm{Coh}X)$ has a Serre functor by using the fact that $Hom(A,-)^*$ is representable for every bounded complex of coherent sheaves $A$.